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A primer of the complex WKB method, with application to the ODE/IM correspondence

Gabriele Degano, Davide Masoero

TL;DR

This work develops a rigorous complex WKB framework for a class of anharmonic oscillators in the ODE/IM context, formulating a reduced potential $V(x;E,\ell)=x^{2\alpha}-E+\frac{(\ell+\tfrac{1}{2})^2}{x^2}$ and constructing WKB approximants via $\Psi^W$ along admissible curves. It proves a Fundamental Theorem on curves that yields a controlled, unique WKB solution through a Volterra equation, and extends to global asymptotics using quadratic differentials, Stokes phenomena, asymptotic values, and Fock-Goncharov coordinates, culminating in Bohr-Sommerfeld quantisation for the spectrum. The paper then derives two principal asymptotic regimes for the spectral problem: (i) large energy with fixed $\ell$, and (ii) large energy and angular momentum with a rescaled parameter $\nu$, providing precise error bounds and connections to the spectral determinant $Q_+(E,\ell)$ and to the ODE/IM correspondence. Overall, it offers a self-contained, rigorous pathway from local WKB approximations to global spectral asymptotics, with explicit formulas and proofs that clarify the complex-analytic structure of the problem.

Abstract

In these lectures, we provide an introduction to the complex WKB method, using as a guiding example a class of anharmonic oscillators that appears in the ODE/IM correspondence. In the first three lectures, we introduce the main objects of the method, such as the WKB function, the integral equations of Volterra type, the quadratic differential and its horizontal/Stokes lines, the Stokes phenomenon, the notion of asymptotic values, the Fock-Goncharov coordinates and their WKB approximation. In the fourth and last lecture, we compute (and prove) the asymptotic behaviour of the spectrum of the anharmonic oscillators in two asymptotic regimes, when the momentum is fixed and the energy is large, and when the momentum (hence also the energy) is large.

A primer of the complex WKB method, with application to the ODE/IM correspondence

TL;DR

This work develops a rigorous complex WKB framework for a class of anharmonic oscillators in the ODE/IM context, formulating a reduced potential and constructing WKB approximants via along admissible curves. It proves a Fundamental Theorem on curves that yields a controlled, unique WKB solution through a Volterra equation, and extends to global asymptotics using quadratic differentials, Stokes phenomena, asymptotic values, and Fock-Goncharov coordinates, culminating in Bohr-Sommerfeld quantisation for the spectrum. The paper then derives two principal asymptotic regimes for the spectral problem: (i) large energy with fixed , and (ii) large energy and angular momentum with a rescaled parameter , providing precise error bounds and connections to the spectral determinant and to the ODE/IM correspondence. Overall, it offers a self-contained, rigorous pathway from local WKB approximations to global spectral asymptotics, with explicit formulas and proofs that clarify the complex-analytic structure of the problem.

Abstract

In these lectures, we provide an introduction to the complex WKB method, using as a guiding example a class of anharmonic oscillators that appears in the ODE/IM correspondence. In the first three lectures, we introduce the main objects of the method, such as the WKB function, the integral equations of Volterra type, the quadratic differential and its horizontal/Stokes lines, the Stokes phenomenon, the notion of asymptotic values, the Fock-Goncharov coordinates and their WKB approximation. In the fourth and last lecture, we compute (and prove) the asymptotic behaviour of the spectrum of the anharmonic oscillators in two asymptotic regimes, when the momentum is fixed and the energy is large, and when the momentum (hence also the energy) is large.
Paper Structure (21 sections, 47 theorems, 206 equations, 14 figures)

This paper contains 21 sections, 47 theorems, 206 equations, 14 figures.

Table of Contents

  1. Lecture I. Fundamental Theorem of the WKB approximation
  2. WKB functions for the anharmonic oscillator
  3. Exercises. I
  4. Lecture II. Asymptotic behaviour of solutions - Stokes Phenomenon
  5. Quadratic differential
  6. Behaviour of solutions at $\infty$: Stokes phenomenon
  7. Behaviour of solutions at $0$
  8. Fock-Goncharov Coordinates
  9. Exercises. II
  10. Lecture III. WKB approximation of Fock-Goncharov co-ordinates
  11. From a quadrilateral to a cycle through surgery
  12. Stokes Complex
  13. Case I: $\mathbf{0\leq E < E_*=2 \ell+1}$
  14. $\mathbf{E = E_*=2 \ell+1}$
  15. $\mathbf{E > E_*=2 \ell+1}$
  16. ...and 6 more sections

Key Result

Lemma 1.1

Let $D\subset \mathbb{C}^*$ be a domain, $\gamma\colon (0,1) \to D$ be an injective smooth curve. Assume that the following two conditions hold with $F(\cdot), B(\cdot,\cdot)$ as in (eq:forcing, eq:integral), and $\dot{\gamma}(t)=\frac{d}{dt}\gamma(t)$. If the equation eq:schr admits a solution $\psi(x;\underline{u})$ such that then the ratio $z(x):=\frac{\psi(x)}{\Psi(x)}$, restricted to $\gamm

Figures (14)

  • Figure 1: Horizontal trajectories in a neighbourhood of a regular point and in a neighbourhood of $\infty$ when $\alpha=\frac{1}{2}$.
  • Figure 2: $2\alpha \in \mathbb{N}$. Horizontal trajectories in a neighbourhood of $x=0$, when $\operatorname{Im}\left(\ell+\frac{1}{2}\right)=0$, $\operatorname{Re}\left(\ell+\frac{1}{2}\right)=0$, and when $\operatorname{Im}\left(\ell+\frac{1}{2}\right),\operatorname{Re}\left(\ell+\frac{1}{2}\right)\neq 0$.
  • Figure 3: Topological representation of a quadruple of admissible lines joining the vertices of an admissible quadrilateral $x_ax_bx_cx_d$ before and after the surgery.
  • Figure 4: Horizontal trajectories in a neighbourhood of a simple turning point.
  • Figure 5: Isotropic harmonic oscillator potential with $\ell>-\frac{1}{2}$ and $E<2\ell+1$. Left: Topological representation of the Stokes complex. Right: Admissibility graph.
  • ...and 9 more figures

Theorems & Definitions (121)

  • Definition 1
  • Remark 2
  • Remark 3
  • Lemma 1.1
  • Definition 1.2
  • Definition 1.3
  • Proposition 1.4
  • proof
  • Corollary 1.5
  • proof
  • ...and 111 more